1. **State the problem:** Solve the equation $$2 \frac{1}{4} x - 3 \frac{1}{2} = -\frac{1}{16}$$ for $x$. 2. **Convert mixed numbers to improper fractions:** $$2 \frac{1}{4} = \frac{9}{4}, \quad 3 \frac{1}{2} = \frac{7}{2}$$ 3. **Rewrite the equation:** $$\frac{9}{4} x - \frac{7}{2} = -\frac{1}{16}$$ 4. **Isolate the term with $x$ by adding $\frac{7}{2}$ to both sides:** $$\frac{9}{4} x = -\frac{1}{16} + \frac{7}{2}$$ 5. **Find common denominator for the right side:** $$\frac{7}{2} = \frac{56}{16}$$ 6. **Add the fractions:** $$-\frac{1}{16} + \frac{56}{16} = \frac{55}{16}$$ 7. **So,** $$\frac{9}{4} x = \frac{55}{16}$$ 8. **Solve for $x$ by dividing both sides by $\frac{9}{4}$:** $$x = \frac{55}{16} \div \frac{9}{4} = \frac{55}{16} \times \frac{4}{9}$$ 9. **Simplify the multiplication:** $$x = \frac{55 \times 4}{16 \times 9} = \frac{220}{144}$$ 10. **Simplify the fraction by dividing numerator and denominator by 4:** $$x = \frac{\cancel{220}^ {55}}{\cancel{144}^{36}}$$ 11. **Final simplified answer:** $$x = \frac{55}{36}$$ **Answer:** $x = \frac{55}{36}$